Problem: Michael is 3 times as old as Emily. Six years ago, Michael was 5 times as old as Emily. How old is Emily now?
We can use the given information to write down two equations that describe the ages of Michael and Emily. Let Michael's current age be $m$ and Emily's current age be $e$ The information in the first sentence can be expressed in the following equation: $m = 3e$ Six years ago, Michael was $m - 6$ years old, and Emily was $e - 6$ years old. The information in the second sentence can be expressed in the following equation: $m - 6 = 5(e - 6)$ Now we have two independent equations, and we can solve for our two unknowns. Because we are looking for $e$ , it might be easiest to use our first equation for $m$ and substitute it into our second equation. Our first equation is: $m = 3e$ . Substituting this into our second equation, we get: $3e$ $-$ $6 = 5(e - 6)$ which combines the information about $e$ from both of our original equations. Simplifying the right side of this equation, we get: $3 e - 6 = 5 e - 30$ Solving for $e$ , we get: $2 e = 24.$ $e = 12$.